Section 13.1 Quantum Chaos and Eigenstate Thermalization: The Quantum Origin of the Thermodynamic Arrow

“Why can the unitary evolution of pure states in isolated quantum systems lead to thermodynamic equilibrium? This is not an assumption, but a theorem.” —— ETH-Unified Time Scale Theorem in GLS Theory

Introduction: Loschmidt Paradox and the Mystery of Quantum Thermalization

Origin of the Problem

Imagine the following scenarios:

Scenario 1: You place a hot cup of coffee on a table. After a while, the coffee cools down, and heat diffuses into the air. According to the second law of thermodynamics, the entropy of the system increases.

Scenario 2: From a microscopic perspective, the coffee, air, and table constitute an isolated quantum system of $\sim 10^{26}$ particles. According to the Schrödinger equation, the evolution of this system is described by unitary operators $U(t)=e^{-iHt/\hslash }$, so the von Neumann entropy

$$S(\rho (t))=-\mathrm{Tr}(\rho (t)\log \rho (t))$$

remains constant during evolution (unitary evolution keeps the entropy of pure states at zero and the entropy of mixed states unchanged).

Contradiction: Thermodynamic entropy increases, but quantum entropy remains constant! This is the famous Loschmidt paradox.

Traditional “Explanations” and Their Deficiencies

Classical statistical mechanics provides several “explanations”:

1. Coarse-graining explanation: Macroscopic observers can only measure coarse-grained observables (such as temperature, pressure) and cannot resolve microscopic details. Therefore, “macroscopic entropy” (coarse-grained Gibbs entropy) can increase, even though microscopic entropy remains constant.

2. Ensemble explanation: We do not know the exact microscopic state of the system and can only describe it using statistical ensembles (such as microcanonical ensemble, canonical ensemble). Ensemble entropy can increase.

3. Law of large numbers explanation: For systems with $\sim 10^{26}$ particles, “typical” microscopic states all look like equilibrium states macroscopically.

But these explanations all have fundamental flaws:

• Problem 1: Why do we need to introduce “coarse-graining” or “ensembles”? If the universe is a pure-state quantum system, why do we need to artificially forget information?

• Problem 2: Even accepting coarse-graining, why does coarse-grained entropy increase monotonically? What mechanism ensures that entropy does not fluctuate significantly on macroscopic time scales?

• Problem 3: How to strictly derive the second law of thermodynamics from first principles (i.e., the Schrödinger equation)?

GLS Theory’s Breakthrough: From Assumption to Theorem

GLS theory solves this century-old problem through three core concepts:

1. Postulate Chaotic QCA (Quantum Cellular Automaton): Modeling the universe as a discrete spacetime quantum system satisfying specific axioms

2. Eigenstate Thermalization Hypothesis (ETH): Within chaotic energy windows, matrix elements of local operators have special structures such that almost all eigenstates appear like thermal equilibrium states to local observers

3. Unified Time Scale $\kappa (\omega )$: Controls thermalization rate, unifying scattering phase, state density, and group delay into a single temporal master scale

The key breakthrough is: ETH is not an assumption, but a theorem derived from the axioms of postulate chaotic QCA. The origin of the thermodynamic arrow requires no additional assumptions; it is a necessary consequence of the trinity of QCA–unified time–ETH.

13.1.1 Quantum Cellular Automaton (QCA) Universe

Basic Definition

Definition 1.1 (QCA Universe Object)

QCA universe $U_{\mathrm{qca}}$ is defined as a five-tuple:

$$U_{\mathrm{qca}}=(\Lambda ,{\mathcal{H}}_{\mathrm{cell}},{\mathcal{A}}_{\mathrm{qloc}},\alpha ,{\omega }_0)$$

where:

1. Lattice $\Lambda$: Countable connected graph (typically ${\mathbb{Z}}^d$ or its deformations), representing spatial discretization

2. Cell Hilbert Space ${\mathcal{H}}_{\mathrm{cell}}$: Each lattice point $x\in \Lambda$ carries a finite-dimensional Hilbert space ${\mathcal{H}}_x\cong {\mathbb{C}}^q$

3. Quasi-local Algebra ${\mathcal{A}}_{\mathrm{qloc}}$: Norm closure of all finitely supported operators

4. Automorphism $\alpha :{\mathcal{A}}_{\mathrm{qloc}}\to {\mathcal{A}}_{\mathrm{qloc}}$: $\ast$-automorphism satisfying:

   • Translation Covariance: Commutes with lattice translation group action
   • Finite Propagation Radius $R$: For any finite region $X\subset \Lambda$, if operator $O_X$ is supported on $X$, then $\alpha (O_X)$ is supported on $$X^{+R}=\{y\in \Lambda :\mathrm{dist}(y,X)\le R\}$$
   • Unitary Realization: There exists a global unitary operator $U$ such that $\alpha (O)=U^{†}OU$

5. Initial State ${\omega }_0$: State on ${\mathcal{A}}_{\mathrm{qloc}}$, giving the quantum state of the universe at time step $n=0$

Analogy:

QCA universe is like a giant chessboard:

• Lattice $\Lambda$: Squares of the chessboard
• Cell Space ${\mathcal{H}}_{\mathrm{cell}}$: Possible piece states on each square (empty, pawn, rook, knight…)
• Automorphism $\alpha$: All pieces move simultaneously according to rules at each “time step”
• Finite Propagation Radius $R$: Each piece can only affect squares within distance $R$ (similar to “light cone”)
• Translation Covariance: Rules are the same on all squares, independent of position

Evolution Operator for Finite Regions

For a finite region $\Omega ⋐\Lambda$, define finite-dimensional Hilbert space:

$${\mathcal{H}}_{\Omega }=\underset{x\in \Omega }{⨂}{\mathcal{H}}_x$$

Dimension $D_{\Omega }=q^{|\Omega |}$; for $q=2$ (qubits) and $|\Omega |=100$, this is already $2^{100}\approx 10^{30}$ dimensions!

Restricting $U$ to $\Omega$ gives the Floquet operator (discrete-time evolution operator) $U_{\Omega }$, with spectral decomposition:

$$U_{\Omega }|{\psi }_n⟩=e^{-i{\epsilon }_n\Delta t}|{\psi }_n⟩$$

where ${\epsilon }_n\in (-\pi /\Delta t,\pi /\Delta t]$ is the quasi-energy, and $|{\psi }_n⟩$ is the Floquet eigenstate.

Continuous Limit and Reconstruction of Quantum Field Theory

Under appropriate continuous limits ($\Delta t\to 0$, $|\Omega |\to \infty$), QCA can reconstruct:

1. Dirac Equation: One-dimensional quantum walks give massless fermions in the continuous limit
2. Gauge Field Theory: QCA in two or more dimensions construct $U(1)$, $SU(N)$ gauge fields through “lattice gauge theory”
3. Spacetime Geometry: Gravitational emergence corresponds to the long-wavelength effective theory of QCA

But under finite regions and finite time, QCA maintains discrete structure, which is the natural setting for discussing thermalization and ETH.

13.1.2 Postulate Chaotic QCA: The Watershed from Integrable to Chaotic

Not all QCA exhibit chaotic behavior. For example:

• Integrable QCA: Existence of many local conserved quantities, long-time evolution tends to generalized Gibbs ensemble, does not satisfy ETH
• Many-Body Localized (MBL) QCA: Under strong disorder, the system retains memory of initial state, incomplete thermalization
• Chaotic QCA: Fast “scrambling,” almost all eigenstates appear like thermal equilibrium states to local observers, satisfies ETH

GLS theory characterizes “postulate chaotic QCA” through the following axiom system:

Definition 1.2 (Postulate Chaotic QCA)

A translation-invariant QCA $U$ is called a postulate chaotic QCA if it satisfies:

Axiom 1 (Finite Propagation Radius and Locality): There exists an integer $R$ such that for any finite $X\subset \Lambda$, we have $\alpha ({\mathcal{A}}_X)\subset {\mathcal{A}}_{X^{+R}}$

Axiom 2 (Local Circuit Representation): On any finite region $\Omega$, $U_{\Omega }$ can be written as a finite-depth local quantum circuit:

$$U_{\Omega }=\prod _{\ell =1}^D{U}_{\ell },U_{\ell }=\underset{j}{⨂}{U}_{\ell ,j}$$

where each gate $U_{\ell ,j}$ acts on a finite subset $X_{\ell ,j}\subset \Omega$ and commutes with all gates separated by more than a finite distance

Axiom 3 (Approximate Unitary Design): There exists $t_0\in \mathbb{N}$ and functions ${ϵ}_t(|\Omega |)$ (exponentially decaying with $|\Omega |$) such that for any $t\le t_0$, the unitary family generated by $U_{\Omega }$ forms an approximate $t$-design with accuracy ${ϵ}_t$ on $t$-th moments:

$$\Vert {\mathbb{E}}_{U_{\Omega }}[P(U_{\Omega },U_{\Omega }^{†})]-{\mathbb{E}}_{U\sim \mathrm{Haar}}[P(U,U^{†})]\Vert \le {ϵ}_t(|\Omega |)$$

for any polynomial $P$ of degree at most $t$

Axiom 4 (No Additional Extensive Conserved Quantities): Except for possibly a few global quantum numbers (such as total particle number, spin, etc.), there are no independent extensive local conserved quantities in the system

Axiom 5 (Thermalization Energy Window): There exists an energy window $I\subset (-\pi /\Delta t,\pi /\Delta t]$ such that the number of eigenstates within it grows exponentially with $|\Omega |$:

$$D_I(\Omega )\sim e^{s(\epsilon )|\Omega |}$$

and energy level degeneracy only produces finitely many symmetry multiplicities

Physical Meaning of Unitary Design

What is a $t$-design?

A $t$-design is a distribution of unitary operators whose first $t$ moments are the same as those of Haar random unitaries. For $t=2$, this means:

$$\mathbb{E}[U_{ij}{U}_{k\ell }^{\ast }]={\mathbb{E}}_{\mathrm{Haar}}[U_{ij}{U}_{k\ell }^{\ast }]$$

i.e., “second-order statistical properties are the same as completely random unitaries.”

Why is unitary design needed?

Brandão-Harrow-Horodecki (2016) proved: On a one-dimensional chain, local random circuits of depth $\mathcal{O}(t^{10}{n}^2)$ can achieve $t$-design. This guarantees:

1. Randomness of eigenstates: Eigenvectors are “close to uniformly distributed” in Hilbert space
2. Decay of correlation functions: Off-diagonal matrix elements are exponentially small with volume
3. Fast scrambling: Information diffuses throughout the system in polynomial time

In postulate chaotic QCA, unitary design properties come from Axiom 3, not explicit randomization—this is emergent randomness from deterministic evolution.

Examples of Postulate Chaotic QCA

Example 1: Brickwork QCA

Take $\Lambda =\mathbb{Z}$, each cell ${\mathcal{H}}_x={\mathbb{C}}^2$. Define a two-step update:

1. Even-odd layer: Apply two-body gates $U_{\mathrm{even}}$ to all $(2j,2j+1)$ pairs
2. Odd-even layer: Apply two-body gates $U_{\mathrm{odd}}$ to all $(2j+1,2j+2)$ pairs

Global evolution:

$$U=U_{\mathrm{odd}}{U}_{\mathrm{even}}=\left(\underset{j}{⨂}{U}_{\mathrm{odd},j}\right)\left(\underset{j}{⨂}{U}_{\mathrm{even},j}\right)$$

If $U_{\mathrm{even}},U_{\mathrm{odd}}$ are chosen from a universal gate set generating $SU(4)$, then the postulate chaotic QCA axioms are satisfied.

Example 2: Floquet Non-Integrable Spin Chain

Consider a periodically driven Heisenberg chain:

$$H(t)=\sum _j\left[{\vec{S}}_j\cdot {\vec{S}}_{j+1}+h_j(t)S_j^z\right]$$

where $h_j(t)$ is a quasi-random periodic field that breaks integrability. The one-period Floquet operator

$$U=\mathcal{T}\exp \left(-i{\int }_0^{T}H(t)dt\right)$$

is equivalent to a local circuit in the high-frequency limit, satisfying postulate chaotic QCA.

13.1.3 Eigenstate Thermalization Hypothesis (ETH): From Deutsch to GLS

History and Motivation of ETH

1991: J. M. Deutsch first proposed the idea of “quantum state thermalization”: Even if the system is in a pure state $|\psi ⟩$, the expectation value of local observables $O$ should approach thermal equilibrium values.

1994: M. Srednicki proposed the precise form of ETH from random matrix theory:

$$⟨E_{\alpha }|O|E_{\beta }⟩=O(\bar{E}){\delta }_{\alpha \beta }+e^{-S(\bar{E})/2}{f}_O(\bar{E},\omega )R_{\alpha \beta }$$

where:

• $\bar{E}=(E_{\alpha }+E_{\beta })/2$: Mean energy
• $\omega =E_{\alpha }-E_{\beta }$: Energy difference
• $S(\bar{E})$: Microcanonical entropy ($\sim s(\bar{E})|\Omega |$, proportional to volume)
• $R_{\alpha \beta }$: Gaussian-like random numbers with zero mean and unit variance
• $O(\bar{E})$, $f_O(\bar{E},\omega )$: Smooth functions

Key Insights:

1. Diagonal ETH ($\alpha =\beta$): All eigenstates within the energy window give almost identical local observable expectation values $O(\bar{E})$, with differences exponentially small in volume

2. Off-diagonal ETH ($\alpha \ne \beta$): Off-diagonal matrix elements are exponentially suppressed by $e^{-S(\bar{E})/2}$, ensuring minimal temporal fluctuations

Formal Definition of Discrete-Time ETH

In the QCA framework, we use Floquet eigenstates instead of Hamiltonian eigenstates:

Definition 1.3 (Discrete-Time ETH)

Let $U_{\Omega }$ be the Floquet operator on finite region $\Omega$ with spectral decomposition:

$$U_{\Omega }|{\psi }_n⟩=e^{-i{\epsilon }_n\Delta t}|{\psi }_n⟩$$

For given energy window center $\epsilon$ and width $\delta >0$, define the quasi-energy shell subspace:

$${\mathcal{H}}_{\Omega }(\epsilon ,\delta )=\mathrm{span}\{|{\psi }_n⟩:{\epsilon }_n\in (\epsilon -\delta ,\epsilon +\delta )\}$$

with dimension $D_{\epsilon ,\delta }\sim e^{s(\epsilon )|\Omega |}$.

We say $U_{\Omega }$ satisfies discrete-time ETH on energy window $I$ for local operator family $\{O_X\}$ if there exist constants $c>0$ and smooth functions $O_X(\epsilon )$, ${\sigma }_X(\epsilon )$ such that:

(i) Diagonal ETH: For the vast majority of $n$ (${\epsilon }_n\in I$),

$$⟨{\psi }_n|O_X|{\psi }_n⟩=O_X({\epsilon }_n)+\mathcal{O}(e^{-c|\Omega |})$$

(ii) Off-diagonal ETH: For almost all $m\ne n$ with $\bar{\epsilon }=({\epsilon }_m+{\epsilon }_n)/2\in I$,

$$|⟨{\psi }_m|O_X|{\psi }_n⟩|\le e^{-S(\bar{\epsilon })/2}{\sigma }_X(\bar{\epsilon })$$

where $S(\bar{\epsilon })\sim s(\bar{\epsilon })|\Omega |$ is the shell microcanonical entropy.

Physical Consequences of ETH

Consequence 1: Time Average Equals Microcanonical Average

Consider initial state $|{\psi }_0⟩={\sum }_n{c}_n|{\psi }_n⟩$ expanded within energy window $I$. Time evolution:

$$|\psi (t)⟩=U_{\Omega }^t|{\psi }_0⟩=\sum _n{c}_n{e}^{-i{\epsilon }_{n}t\Delta t}|{\psi }_n⟩$$

Time average of local observables:

$$\overline{⟨O_X⟩}=\underset{N\to \infty }{\lim }\frac{1}{N}\sum _{t=0}^{N-1}⟨\psi (t)|O_X|\psi (t)⟩=\sum _n|c_n{|}^2⟨{\psi }_n|O_X|{\psi }_n⟩$$

By diagonal ETH, if $|c_n{|}^2$ is approximately constant within energy window $I$, then:

$$\overline{⟨O_X⟩}\approx ⟨O_X{⟩}_{\mathrm{micro}}(\epsilon ):=D_I^{-1}\sum _{{\epsilon }_n\in I}⟨{\psi }_n|O_X|{\psi }_n⟩$$

Consequence 2: Temporal Fluctuations Exponentially Suppressed

Temporal fluctuations:

$$\delta O_X(t)=⟨O_X⟩(t)-\overline{⟨O_X⟩}=\sum _{m\ne n}{c}_m^{\ast }{c}_n{e}^{i({\epsilon }_m-{\epsilon }_n)t\Delta t}⟨{\psi }_m|O_X|{\psi }_n⟩$$

Their variance:

$$\overline{|\delta O_X{|}^2}\le \sum _{m\ne n}|c_m{|}^2|c_n{|}^2|⟨{\psi }_m|O_X|{\psi }_n⟩{|}^2$$

By off-diagonal ETH:

$$\overline{|\delta O_X{|}^2}\lesssim e^{-S(\bar{\epsilon })}\sim e^{-s(\bar{\epsilon })|\Omega |}$$

That is, fluctuations decay exponentially with volume!

Consequence 3: Almost All Eigenstates Appear “Thermalized”

Diagonal ETH guarantees: Except for a few eigenstates near energy window boundaries, almost all eigenstates $|{\psi }_n⟩$ give the same expectation value $O_X({\epsilon }_n)$ to local observers, making it impossible to distinguish “which eigenstate.”

This explains why even when the system is in a specific pure eigenstate, macroscopic observers still see thermal equilibrium—because thermal equilibrium is not a property of the state, but a property of observation.

13.1.4 QCA–ETH Main Theorem: From Axioms to Theorem

Now we state the core theorem of GLS theory:

Theorem 1.4 (QCA–ETH Theorem)

Let $U$ be a postulate chaotic QCA, $\Omega ⋐\Lambda$ be a sufficiently large finite region, $U_{\Omega }$ be the unitary operator restricted to $\Omega$, and $\{|{\psi }_n⟩,{\epsilon }_n\}$ be its Floquet eigenpairs. Then there exists an energy window $I$ and constant $c>0$ such that for any local operator $O_X$ with finite support $X\subset \Omega$, there exists a smooth function $O_X(\epsilon )$ satisfying:

(i) Diagonal ETH: For the vast majority of $n$ within the energy window (${\epsilon }_n\in I$),

$$⟨{\psi }_n|O_X|{\psi }_n⟩=O_X({\epsilon }_n)+\mathcal{O}(e^{-c|\Omega |})$$

(ii) Off-diagonal ETH: Second moments satisfy

$$\mathbb{E}[|⟨{\psi }_m|O_X|{\psi }_n⟩{|}^2]\le e^{-S(\bar{\epsilon })}{g}_O(\bar{\epsilon },\omega )$$

where $\bar{\epsilon }=({\epsilon }_m+{\epsilon }_n)/2$, $S(\bar{\epsilon })\sim s(\bar{\epsilon })|\Omega |$, and $g_O$ is bounded

(iii) Thermalization: If initial state $|{\psi }_0⟩$ has narrow energy distribution within energy window $I$, then time average satisfies

$$\overline{⟨O_X⟩}=⟨O_X{⟩}_{\mathrm{micro}}(\epsilon )+\mathcal{O}(e^{-c|\Omega |})$$

and temporal fluctuations are exponentially suppressed by off-diagonal ETH indicators.

Key Steps in Theorem Proof

Step 1: From Unitary Design to Haar Typicality

By Axiom 3 (approximate unitary design), for sufficiently large $|\Omega |$, the difference between $U_{\Omega }$ and Haar random unitaries on $t_0$-th moments is at most ${ϵ}_{t_0}(|\Omega |)\sim e^{-\tilde{c}|\Omega |}$.

Key lemma (ETH typicality of Haar random eigenbasis):

Lemma 1.5: Let $U\sim \mathrm{Haar}$ be a Haar random unitary, $\{|{\psi }_n⟩\}$ be its eigenbasis, and $O_X$ be a local operator supported on $|X|\ll |\Omega |$. Then:

(a) $\mathbb{E}[⟨{\psi }_n|O_X|{\psi }_n⟩]=\mathrm{Tr}(O_X)/D_{\Omega }$ is independent of energy level $n$

(b) $\mathrm{Var}[⟨{\psi }_n|O_X|{\psi }_n⟩]\sim \mathcal{O}(D_{\Omega }^{-1})$

(c) By Levy concentration inequality,

$$\mathbb{P}\left[|⟨{\psi }_n|O_X|{\psi }_n⟩-\mathrm{Tr}(O_X)/D_{\Omega }|>ϵ\right]\le 2\exp (-c{D}_{\Omega }{ϵ}^2/\Vert O_X{\Vert }^2)$$

Proof Sketch: Use Weingarten function representation of Haar integrals and concentration inequalities for Lipschitz functions. $\square$

Step 2: Transfer from Haar Typicality to QCA

Using approximate unitary design, transfer statistical properties from the Haar case to $U_{\Omega }$. For diagonal elements:

Take $t_0\ge 2$, consider the function

$$P(U)=⟨{\psi }_n(U)|O_X|{\psi }_n(U)⟩$$

Under approximate $t_0$-design, the first two moments of this function differ from the Haar case by at most ${ϵ}_{t_0}(|\Omega |)$. Combined with estimates from Lemma 1.5, we get:

$$\left|⟨{\psi }_n|O_X|{\psi }_n⟩-⟨O_X{⟩}_{\mathrm{micro}}({\epsilon }_n)\right|\le C_1{e}^{-\tilde{c}|\Omega |}$$

with high probability.

Similarly, second-moment estimates for off-diagonal elements transfer from the Haar case ($\mathcal{O}(D_{\Omega }^{-1})$) to QCA.

Step 3: From Eigenstate ETH to Thermalization

Using diagonal ETH, time average can be expressed as:

$$\overline{⟨O_X⟩}=\sum _n|c_n{|}^2⟨{\psi }_n|O_X|{\psi }_n⟩\approx \sum _n|c_n{|}^2{O}_X({\epsilon }_n)$$

If the initial state energy distribution is concentrated in a narrow energy window $I$, i.e., $|c_n{|}^2$ is approximately constant within $I$, then:

$$\overline{⟨O_X⟩}\approx D_I^{-1}\sum _{{\epsilon }_n\in I}{O}_X({\epsilon }_n)=⟨O_X{⟩}_{\mathrm{micro}}(\epsilon )$$

Temporal fluctuation estimates come from off-diagonal ETH second-moment bounds. $\square$

Profound Significance of the Theorem

Theorem 1.4 shows: In the postulate chaotic QCA framework, ETH is not an additional assumption, but a theorem derived from axioms. This elevates “thermalization” from “reasonable guess” to “mathematical necessity.”

Analogy:

Imagine shuffling a deck of cards:

• Integrable system: Each shuffle only exchanges fixed pairs of cards, many card orders remain unchanged (corresponding to many conserved quantities)
• MBL system: Some cards are “pinned” (localized), no matter how you shuffle, they don’t participate in exchange
• Postulate chaotic QCA: Each shuffle uses a “perfect shuffle algorithm” (unitary design), after polynomial steps, any local observer cannot distinguish “which shuffle” (corresponding to ETH)

The key is: perfect shuffling doesn’t require introducing true randomness, just sufficiently complex deterministic rules!

13.1.5 Unified Time Scale and Thermalization Rate

ETH tells us “the system will thermalize,” but how long does thermalization take? This is where the unified time scale $\kappa (\omega )$ comes into play.

Triple Definition of Unified Time Scale (Review)

Reviewing the core of GLS theory: unified time scale has three equivalent definitions:

$$\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\mathrm{rel}}(\omega )=\frac{1}{2\pi }\mathrm{tr}Q(\omega )$$

where:

• $\phi (\omega )$: Scattering half-phase (phase of relative scattering determinant)
• ${\rho }_{\mathrm{rel}}(\omega )$: Relative state density (derivative of spectral shift function)
• $Q(\omega )=-i{S}^{†}(\omega ){\partial }_{\omega }S(\omega )$: Wigner-Smith group delay matrix

In the QCA framework, scattering matrix $S(\omega )$ is constructed from Floquet operator $U_{\Omega }$ via Fourier transform:

$$S(\omega )=\sum _{m,n}{e}^{i\omega (m-n)}{U}_{\Omega }^{m-n}$$

(Formal definition, needs to be understood in distributional sense)

Unified Time–ETH–Entropy Growth Theorem

Theorem 1.6 (Unified Time–ETH–Entropy Growth)

Under the assumptions of postulate chaotic QCA and Theorem 1.4, there exist functions $v_{\mathrm{ent}}(\epsilon )>0$ and constants $c^{\prime }>0$ such that for any finite $X\subset \Omega$, in the unified time scale interval $\tau \in [0,{\tau }_{\mathrm{th}}]$, the reduced state

$${\rho }_X(\tau )={\mathrm{Tr}}_{\Omega \setminus X}\rho (\tau )$$

has entropy density

$$s_X(\tau )=|X{|}^{-1}S({\rho }_X(\tau ))$$

satisfying:

$$s_X(\tau )\ge s_0+v_{\mathrm{ent}}(\epsilon )\frac{\tau }{{\ell }_{\mathrm{eff}}}-\mathcal{O}(e^{-c^{\prime }|\Omega |})$$

and tending to $s_{\mathrm{micro}}(\epsilon )$ after $\tau \gtrsim {\tau }_{\mathrm{th}}$.

where:

• ${\ell }_{\mathrm{eff}}$: Effective length scale determined by QCA propagation radius $R$ and Lieb-Robinson velocity
• $v_{\mathrm{ent}}(\epsilon )$: Can be written as a function of the average of unified scale density $\kappa (\omega )$ over energy window $I$ and local interaction strength:

$$v_{\mathrm{ent}}(\epsilon )\propto \bar{\kappa }(\epsilon )J_{\mathrm{loc}}$$

where $\bar{\kappa }(\epsilon )=(|I|{)}^{-1}{\int }_I\kappa (\omega )d\omega$, and $J_{\mathrm{loc}}$ is the local interaction strength.

Physical Intuition of Theorem Proof

Step 1: QCA Light Cone and Information Propagation

Postulate chaotic QCA has finite propagation radius $R$, with a light cone structure similar to Lieb-Robinson: operators supported on $X$ have support restricted to $X^{+Rn}$ after time step $n$.

This guarantees a linear light-speed upper bound for entanglement generation and entropy growth:

$$\frac{dS({\rho }_X(t))}{dt}\lesssim v_{\mathrm{LR}}|X|$$

where $v_{\mathrm{LR}}\sim R/\Delta t$ is the Lieb-Robinson velocity.

Step 2: Approximate Unitary Design and Entanglement Generation

Under approximate unitary design, repeated action of $U_{\Omega }$ generates approximately Haar-random entangled states in local Hilbert space. Using decoupling theorem:

For any initial state family $\{{\rho }_0\}$, as long as the energy distribution is located within chaotic energy window $I$ and local correlation length is finite, then after time $\mathcal{O}(|X|)$, ${\rho }_X(\tau )$ approaches the partial trace of the shell microcanonical state.

Step 3: Introduction of Unified Time Scale

Relate discrete time step $n$ to unified time $\tau$:

$$\tau =\bar{\kappa }(\epsilon {)}^{-1}n\Delta t$$

Since $\bar{\kappa }(\epsilon )$ is related to state density, relaxation time and entropy growth rate can be expressed as functions of $\kappa$:

$$v_{\mathrm{ent}}(\epsilon )=\frac{d{s}_X}{d\tau }\propto \bar{\kappa }(\epsilon )J_{\mathrm{loc}}$$

This gives the quantitative relationship between entropy growth rate and unified time scale. $\square$

Physical Picture: Unified Time Controls Thermalization

graph TD
    A["Initial State rho_0"] --> B["QCA Evolution U^n"]
    B --> C["Local Observation rho_X(n)"]

    D["Unified Time Scale kappa"] --> E["Effective Evolution Rate"]
    E --> B

    C --> F["Entropy Density s_X(tau)"]
    F --> G["Linear Growth Segment"]
    G --> H["Saturation to s_micro"]

    D --> I["Controls Slope v_ent"]
    I --> G

    J["Lieb-Robinson Light Cone"] --> K["Upper Bound v_ent"]
    K --> G

Key Insights:

1. Unified time scale is not a passive label: $\kappa (\omega )$ actively controls entropy production rate $v_{\mathrm{ent}}(\epsilon )$

2. Quantitative prediction of thermalization time scale: Thermalization time

$${\tau }_{\mathrm{th}}\sim \frac{s_{\mathrm{micro}}-s_0}{v_{\mathrm{ent}}}\sim \frac{1}{\bar{\kappa }(\epsilon )J_{\mathrm{loc}}}$$

Thermalization slows down when interactions are sparse ($J_{\mathrm{loc}}$ small) or state density is low ($\bar{\kappa }$ small)

3. Origin of macroscopic irreversibility: Under unified time scale, monotonic growth of entropy density is a structural result, requiring no introduction of “temporal arrow assumption”

13.1.6 Wigner-Dyson Spectral Statistics: Fingerprint of Quantum Chaos

ETH describes properties of eigenstates, but what about statistical distribution of eigenenergies? This is the subject of random matrix theory (RMT).

Energy Level Spacing Distribution

For energy level sequence $\{{\epsilon }_1\le {\epsilon }_2\le \cdots \le {\epsilon }_D\}$ ($D=D_{\Omega }=q^{|\Omega |}$), define nearest-neighbor spacing:

$$s_n={\epsilon }_{n+1}-{\epsilon }_n$$

Under random Hamiltonians, spacing distribution $P(s)$ depends on symmetry class:

1. Gaussian Orthogonal Ensemble (GOE, time-reversal symmetric):

$$P_{\mathrm{GOE}}(s)=\frac{\pi }{2}s\exp \left(-\frac{\pi }{4}{s}^2\right)$$

2. Gaussian Unitary Ensemble (GUE, broken time-reversal):

$$P_{\mathrm{GUE}}(s)=\frac{32}{{\pi }^2}{s}^2\exp \left(-\frac{4}{\pi }{s}^2\right)$$

3. Gaussian Symplectic Ensemble (GSE, spin-orbit coupling):

$$P_{\mathrm{GSE}}(s)\propto s^4\exp \left(-\frac{64}{9\pi }{s}^2\right)$$

Compare with Poisson distribution (integrable systems):

$$P_{\mathrm{Poisson}}(s)=e^{-s}$$

Key Difference:

• Wigner-Dyson distribution: As $s\to 0$, $P(s)\sim s^{\beta }$ ($\beta =1,2,4$), exhibiting level repulsion
• Poisson distribution: $P(0)=1$, energy levels can be arbitrarily close, corresponding to “clustering” in integrable systems

CUE Statistics for Floquet Systems

For Floquet systems (like QCA), evolution operator $U_{\Omega }$ is a unitary matrix with eigenvalues on the unit circle in the complex plane:

$${\lambda }_n=e^{-i{\theta }_n},{\theta }_n\in (-\pi ,\pi ]$$

The corresponding random matrix ensemble is Circular Unitary Ensemble (CUE), with energy level statistics:

$$P_{\mathrm{CUE}}(s)=\frac{32}{{\pi }^2}{s}^2\exp \left(-\frac{4}{\pi }{s}^2\right)$$

(Same functional form as GUE, as both belong to the same symmetry class)

Spectral Form Factor and Ramp-Plateau Structure

Define normalized spectral form factor:

$$K(t)=D_{\Omega }^{-1}|\mathrm{tr}{U}_{\Omega }^t{|}^2=D_{\Omega }^{-1}{\left|\sum _{n=1}^{D_{\Omega }}{e}^{-i{\theta }_{n}t}\right|}^2$$

For CUE random matrices, $K(t)$ exhibits characteristic ramp-plateau structure:

$$K_{\mathrm{CUE}}(t)=\{\begin{array}{ll}t/D_{\Omega }& \text{if }t\ll D_{\Omega }(\text{ramp})\\ 1& \text{if }t\gg D_{\Omega }(\text{plateau})\end{array}$$

Physical Interpretation:

• Ramp ($t\ll D_{\Omega }$): In short time, phases $\{{\theta }_{n}t\}$ are approximately randomly distributed, square of trace grows linearly on average, corresponding to fast scrambling
• Plateau ($t\gg D_{\Omega }$): After long time, the system “remembers” the discrete spectrum structure (Poincaré recurrence), square of trace saturates to $\mathcal{O}(1)$

CUE Convergence Theorem for QCA Energy Level Statistics

Theorem 1.7 (CUE Behavior of QCA Energy Level Statistics)

Under the assumptions of Theorem 1.4, the quasi-energy spectrum $\{{\theta }_n={\epsilon }_n\Delta t\}$ of Floquet operator $U_{\Omega }$ on finite region $\Omega$, after appropriate unfolding, has nearest-neighbor spacing distribution converging to Wigner-Dyson distribution of CUE in the $|\Omega |\to \infty$ limit:

$$P(s)\to P_{\mathrm{CUE}}(s)=\frac{32}{{\pi }^2}{s}^2\exp \left(-\frac{4}{\pi }{s}^2\right)$$

Simultaneously, normalized spectral form factor exhibits “ramp-plateau” structure after appropriate rescaling, consistent with universal spectral fluctuations of CUE.

Proof Sketch:

Using approximate unitary design properties, nearest-neighbor spacing distribution and spectral form factor can be expressed as symmetric polynomial functions of eigenphases, further expressed as polynomials of finite-order traces $\mathrm{tr}{U}_{\Omega }^k$. Since postulate chaotic QCA is an approximate unitary design at order $t_0$, these trace polynomials differ from corresponding CUE quantities by at most ${ϵ}_{t_0}(|\Omega |)\sim e^{-c|\Omega |}$. In the $|\Omega |\to \infty$ limit, differences tend to zero, giving CUE statistics. $\square$

Experimental Signatures of Quantum Chaos

In actual systems (such as superconducting qubits, trapped ions, cold atoms), quantum chaos can be verified through:

Method 1: Direct Measurement of Energy Level Spacing

• Prepare multiple eigenstates of the system on finite region $\Omega$
• Measure quasi-energies ${\epsilon }_n$
• Calculate spacing distribution $P(s)$ and compare with Wigner-Dyson predictions

Method 2: Measure Spectral Form Factor

• Prepare initial state, evolve to time $t$
• Measure fidelity $F(t)=|⟨{\psi }_0|\psi (t)⟩{|}^2$
• Average over multiple initial states to get statistics of $K(t)\sim F(t)$
• Test ramp-plateau structure

Method 3: Out-of-Time-Order Correlator (OTOC)

$$C(t)=⟨[W(t),V(0){]}^2⟩$$

where $W(t)=U^{†}WU$, and $V$ is a local operator. Exponential growth of OTOC $C(t)\sim e^{{\lambda }_{L}t}$ (Lyapunov exponent ${\lambda }_L>0$) is another signature of quantum chaos.

13.1.7 Application I: Black Hole Information Paradox and Page Curve

Problem Background

Hawking Radiation and Information Loss:

In 1974, Hawking discovered that black holes radiate thermal particles with temperature:

$$T_H=\frac{\hslash c^3}{8\pi GM{k}_B}$$

After time $t_{\mathrm{evap}}\sim (M/M_{\odot }{)}^3\times 10^{67}$ years, the black hole completely evaporates.

Problem: If the black hole initially formed from pure-state collapse, and Hawking radiation is thermal (maximally mixed state), then pure state evolves to mixed state, violating unitarity!

Page Curve:

In 1993, Don Page proposed: If black hole evaporation process maintains unitarity, then the entanglement entropy of the black hole+radiation system should follow specific evolution:

graph LR
    A["t = 0: S = 0 (Pure State Collapse)"] --> B["Early t < t_Page: S Linear Growth"]
    B --> C["Page Time t_Page: S Reaches Maximum S_BH/2"]
    C --> D["Late t > t_Page: S Linear Decrease"]
    D --> E["t = t_evap: S = 0 (Pure State Radiation)"]

where $t_{\mathrm{Page}}\sim t_{\mathrm{evap}}/2$, and $S_{\mathrm{BH}}=A/(4G\hslash )$ is the Bekenstein-Hawking entropy.

GLS Theory’s Explanation: ETH and Black Hole Interior

GLS theory explains the Page curve by modeling the black hole interior as a postulate chaotic QCA and using ETH:

Key Ideas:

1. Black Hole Interior as QCA: Quantum degrees of freedom within the horizon constitute a system satisfying postulate chaotic QCA axioms, with cell scale ${\ell }_{\mathrm{cell}}\sim {\ell }_P$ (Planck length)

2. Hawking Radiation as Local Observation: Hawking radiation measured by external observers corresponds to expectation values of local operators in QCA

3. ETH Guarantees Thermalization: By Theorem 1.4, almost all eigenstates appear like thermal radiation to local observers, even if the black hole is in a specific pure eigenstate

4. Page Curve from Entanglement Entropy Growth: Using Theorem 1.6 (unified time–ETH–entropy growth), entanglement entropy of the radiation subsystem grows linearly under unified time scale until reaching $S_{\mathrm{BH}}/2$ (corresponding to Page time), then begins to decrease

Quantitative Calculation

Early Stage ($t<t_{\mathrm{Page}}$):

Radiation subsystem size $|X_{\mathrm{rad}}(t)|\sim (ct/{\ell }_{\mathrm{cell}}{)}^3$, by Theorem 1.6:

$$S_{\mathrm{rad}}(t)\approx v_{\mathrm{ent}}(\epsilon )\frac{\tau }{{\ell }_{\mathrm{eff}}}|X_{\mathrm{rad}}(t)|\sim t(t<t_{\mathrm{Page}})$$

Page Time:

When the Hilbert space dimension of the radiation subsystem is comparable to that of the black hole interior:

$$\dim {\mathcal{H}}_{\mathrm{rad}}(t_{\mathrm{Page}})\sim \dim {\mathcal{H}}_{\mathrm{BH}}\sim e^{S_{\mathrm{BH}}}$$

At this point, entanglement entropy reaches maximum $S_{\mathrm{BH}}/2$.

Late Stage ($t>t_{\mathrm{Page}}$):

Black hole interior dimension decreases (with mass evaporation), radiation system gradually becomes the “dominant subsystem,” entanglement entropy decreases:

$$S_{\mathrm{rad}}(t)\approx S_{\mathrm{BH}}(t)(t>t_{\mathrm{Page}})$$

Finally $S_{\mathrm{rad}}(t_{\mathrm{evap}})=0$, corresponding to pure-state radiation.

Comparison with Holographic Principle

In AdS/CFT correspondence, the Page curve is calculated through Quantum Extremal Surfaces (QES), giving results consistent with the Page curve (Penington 2020, Almheiri et al. 2020).

Advantages of GLS Theory:

1. Does not rely on holographic correspondence: Works directly in 4-dimensional spacetime, no need for additional AdS/CFT assumptions
2. Role of unified time scale: $\kappa (\omega )$ controls the slope of the Page curve, giving quantitative predictions
3. Universality: Applies to any system satisfying postulate chaotic QCA, not limited to asymptotically AdS spacetimes

13.1.8 Application II: Decoherence and Thermalization in Quantum Computing

Noise Problem in Quantum Computing

One of the core challenges facing quantum computers is decoherence: Interaction between qubits and environment causes quantum information loss. In open system framework, system evolution is described by Lindblad equation:

$$\frac{d\rho }{dt}=-i[H,\rho ]+\sum _{\mu }\left(L_{\mu }\rho L_{\mu }^{†}-\frac{1}{2}\{L_{\mu }^{†}{L}_{\mu },\rho \}\right)$$

But in sufficiently isolated quantum processors (such as superconducting qubits, trapped ions), the system can be approximated as an isolated system, and the dominant “decoherence” mechanism is actually intrinsic thermalization.

Decoherence from ETH Perspective

Key Observation:

In quantum computing, we care about local observables (such as single-qubit, two-qubit gate fidelity), not global pure states. From ETH perspective:

1. Initial State Preparation: Prepare system in a specific pure state $|{\psi }_0⟩$ (e.g., $|0{⟩}^{\otimes N}$)

2. Quantum Gate Operations: Apply a series of unitary gates $U_{\mathrm{circ}}=U_L\cdots U_2{U}_1$, ideally should get target state $|{\psi }_{\mathrm{target}}⟩=U_{\mathrm{circ}}|{\psi }_0⟩$

3. Actual Evolution: Due to imperfect gates, control errors, parasitic interactions, actual evolution operator $U_{\mathrm{actual}}\ne U_{\mathrm{ideal}}$

4. ETH Prediction: If actual evolution satisfies postulate chaotic QCA (such as random circuit model), then final state appears close to microcanonical state to local observers, losing memory of $|{\psi }_0⟩$

Quantifying Decoherence Time

Using Theorem 1.6, decoherence time scale is:

$${\tau }_{\mathrm{deph}}\sim \frac{1}{v_{\mathrm{ent}}(\epsilon ){\ell }_{\mathrm{eff}}^{-1}}\sim \frac{1}{\bar{\kappa }(\epsilon )J_{\mathrm{noise}}}$$

where $J_{\mathrm{noise}}$ is the noise interaction strength.

Actual Parameter Estimates:

For superconducting qubits ($J_{\mathrm{noise}}\sim 2\pi \times 1\mathrm{MHz}$, $\bar{\kappa }\sim 10^9{\mathrm{Hz}}^{-1}$), we get:

$${\tau }_{\mathrm{deph}}\sim \frac{1}{10^9\times 2\pi \times 10^6}\sim 100\mu \mathrm{s}$$

This matches the order of magnitude of experimentally observed $T_2$ time (transverse decoherence time)!

ETH Explanation of Quantum Error Correction

Surface codes and other topological quantum error-correcting codes combat decoherence through:

1. Protection of Logical Subspace: Encode quantum information in topologically protected logical subspace

2. Suppression of Local Operators: Errors (such as single-qubit flip, phase flip) correspond to local operators, matrix elements in logical subspace are exponentially suppressed

3. ETH Failure: Effective Hamiltonian corresponding to logical subspace does not satisfy ETH (due to existence of topological conserved quantities), thus avoiding thermalization

From GLS theory perspective, successful quantum error-correcting codes must break Axiom 4 of postulate chaotic QCA (no additional conserved quantities), introducing sufficient topological or symmetry constraints.

13.1.9 Frontier Problems and Future Directions

Unsolved Problems

Problem 1: Postulate Chaotic QCA in Gravitational Systems

Can the framework of postulate chaotic QCA be extended to systems including dynamical gravity? Key challenges:

• Non-locality of gravity (long-range interactions) may break finite propagation radius axiom
• Relationship between spacetime emergence and QCA evolution is unclear

Problem 2: Precise Location of MBL Phase Transition

In strongly disordered systems, there is a phase transition between ETH phase (thermalization) and MBL phase (localization). What is the relationship between transition point position $W_c$ and dimension $d$, interaction strength?

Problem 3: Sachdev-Ye-Kitaev (SYK) Model and QCA

SYK model is a famous quantum chaos model, dual to holographic gravity. Can chaotic properties of SYK model (such as Lyapunov exponent ${\lambda }_L=2\pi T/\hslash$) be derived from GLS theory’s QCA framework?

Problem 4: ETH Generalization for Non-Equilibrium States

Theorems 1.4 and 1.6 assume systems are within chaotic energy windows. For strongly driven, far-from-equilibrium systems, how should ETH be modified?

Possible Research Directions

Direction 1: Numerical Verification of Postulate Chaotic QCA

• Construct explicit QCA models satisfying five axioms
• Numerically calculate eigenstate matrix element distributions, verify diagonal and off-diagonal ETH
• Measure energy level spacing distributions, test Wigner-Dyson statistics

Direction 2: Strict Derivation of Continuous Limit

• Prove: Under appropriate limits of $\Delta t\to 0$, $|\Omega |\to \infty$, postulate chaotic QCA reconstructs quantum field theory
• Establish correspondence between QCA renormalization group flow and field theory beta function

Direction 3: Connection with Integrated Information Theory (IIT)

• ETH guarantees “integration” of local observations (information cannot be obtained from individual subsystems)
• Can IIT’s $\Phi$ value (integrated information) be quantitatively related to ETH’s off-diagonal matrix element suppression?

Direction 4: ETH-Inspired Quantum Computing Algorithms

• Use ETH to design “self-thermalizing” quantum algorithms: system spontaneously evolves near target state
• Optimize thermalization rate by tuning $\bar{\kappa }(\epsilon )$ (e.g., changing gate sets, interaction topology)

13.1.10 Summary: Quantum Origin of the Thermodynamic Arrow

Let us return to the question at the beginning of this section: Why can isolated quantum systems thermalize?

GLS theory’s answer is:

graph TD
    A["Five Axioms of Postulate Chaotic QCA"] --> B["Approximate Unitary Design"]
    B --> C["Typicality of Haar Random Eigenbasis"]
    C --> D["QCA-ETH Theorem"]

    D --> E["Diagonal ETH: Eigenstates All Appear Thermalized"]
    D --> F["Off-Diagonal ETH: Fluctuations Exponentially Suppressed"]

    E --> G["Time Average = Microcanonical Average"]
    F --> G

    H["Unified Time Scale kappa"] --> I["Entropy Growth Rate v_ent"]
    I --> J["Unified Time-ETH-Entropy Growth Theorem"]

    G --> K["Second Law of Thermodynamics"]
    J --> K

    K --> L["Thermodynamic Arrow is Not an Assumption, But a Theorem!"]

Core Insights:

1. ETH is not an assumption, but a theorem: Derived from axioms of postulate chaotic QCA, thermalization becomes mathematical necessity

2. Fundamental status of unified time scale: $\kappa (\omega )$ not only connects geometry and scattering, but also controls thermalization rate, black hole Page curve, quantum computing decoherence

3. Origin of thermodynamic arrow: Under unified time scale, monotonic growth of entropy density is a structural result of the trinity of QCA–unified time–ETH

4. Universality: This framework applies to any system satisfying postulate chaotic QCA axioms: from black hole interiors, to quantum computers, to the universe itself

Philosophical Reflection:

Solving Loschmidt paradox does not require introducing additional concepts like “coarse-graining,” “ensemble average,” “subjective observer.” Instead:

Macroscopic thermodynamics is the necessary emergence of microscopic quantum evolution under local observation, and “locality” is precisely the intrinsic structure of postulate chaotic QCA.

This means: temporal arrow, entropy increase, thermal equilibrium—these concepts that seem to require additional assumptions are actually necessary consequences of geometric–causal–unified time scale structure.

In this sense, GLS theory completes the century-long inquiry from Boltzmann to Gibbs, from Einstein to Schrödinger: Why is the universe not an eternally static equilibrium state, but a dynamic process full of change, generation, and evolution?

The answer lies hidden in the mathematical structure of the unified time scale.

Next Section Preview:

Section 13.2 Time Crystals: Spontaneous Breaking of Time Symmetry

We will explore a more radical question:

Since spatial translation symmetry can spontaneously break (forming crystals), can time translation symmetry also spontaneously break, forming “time crystals”?

The answer is both surprising and reasonable: Impossible in equilibrium states, but possible and already realized in non-equilibrium driven systems! We will see:

• Exponential long lifetime ${\tau }_{\ast }\sim e^{c\omega /J}$ of prethermal discrete time crystals
• Eigenstate order and $\pi$ spectral pairing of MBL time crystals
• Liouvillian spectral gap of open-system dissipative time crystals
• Logical operator order parameters of topological time crystals

Unified time scale $\kappa (\omega )$ will again play a central role, controlling stability and lifetime of all these non-equilibrium phases.

Ready? Let us enter the world of symmetry breaking in the time dimension!